Kevin Kolb found Stephens-Howling on a short toss at the Cardinals' own 48-yard line and he ran through the Cowboys defense all the way to the end zone for the game-winner for Arizona (5-7).
Kolb, who returned to the lineup after missing four games with a turf toe injury, set up the winning play with a 16-yard pass to Andre Roberts, overcoming a false start penalty.
The Cowboys looked like they had the game won in regulation when rookie kicker Dan Bailey connected on a 49-yard field goal as time was running out.
But it was disallowed because Cowboys Coach Jason Garrett had called a timeout before the kick and Bailey's second attempt missed.
Dallas (7-5) saw its four-game winning streak snapped with the overtime setback.
Puzzle-maker slips 'Murdoch Is Evil' into Rupert Murdoch's Sunday Telegraph
Police: Sword-wielding man demanded free tacos